"She only wants half a piece of cake."
Well, then, we have a real conundrum. A piece is a piece, whatever the size. Cut it in half and you have two whole pieces. Then what?
Hahaha good point. However, I think what they were trying to say is "half a slice". (If the cake has already been sliced to slices, and you want just half a slice)
So you slice yourself a slice that's half as large as the original slice? I'm not sure that solves the problem...
But I'm just fooling around.
"想" has to be followed by a verb to mean "to want (to)". In front of a noun it doesn't mean "to want" in the English sense of "to be thinking of having". Instead it means "to miss (someone or something)", "to think up (an idea)", etc. Your sentence makes it sound as though she misses the cake.
Yes, but "我只想要" sounds a little awkward to me. (I'm not sure tho, maybe a native speaker can comment on this?)
KX3 is a native speaker. I imagine you'll get your wish. (Sounds okay to me though.)
It is not as common but still colloquial. I was just a little surprised it wasn't used at all here, but then again their conception of 想、很、要 and such is a little different from what is normal for me and a others here.